Bernard Lang wrote: [about my report on how math works] > Nice field observation report ... > > Now, what do you deduce from it ? > (assuming accuracy and generality) You mean what lessons should we draw from it? The biggest ones have to do with the importance of keeping lines of communication open between different groups if progress is going to happen. The importance of this improves if your endeavor (like with free software) is using a distributed development model. Don't just work out how you will do things. Keep track of how other people are doing then and know how to explain your stuff to them. To name but one example I think that the EROS essays on capabilities are invaluable - I would go so far as to say that they currently have more impact than EROS does! (Visit http://www.eros-os.org/essays/00Essays.html for more.) > - that venture capital should come in and get the field better > organized ? No... > - ... and be rewarded by exclusivity on using results ? No... > - that other fields do not do that (how do you know ?) > and are hence more efficient ? > - ... or is it common in sciences ? ... and in other areas of knowledge My wife's field reports on biology, as well as experiences in other sciences indicate that this situation is specific to math. My wife can pick up most biology papers, certainly anything in molecular and cell biology, and read it. Furthermore in many places, such as the former Eastern Block, this is not a problem. In all cases a clear commitment to keep lines of communication open is key... > (read Sokal) but shows more in math ? > - that math is indeed very difficult ? Anything can be difficult if you make it so. Maintaining lines of communication on technical subjects requires work both on the part of would-be listeners and on the part of the speaker. This effort is worthwhile and failure by either side to recognize this fact is dangerous to the long-term health of your endeavor. > - that it is surprising there there is nevertheless progress in the > field (or no longer) ? It is surprising. There is a lot of activity, but activity is not the same as progress. Plus nobody knows how much is being lost. The question of math being lost is not trivial. For instance I once figured out a nice way to generate answers to such questions as, "Which polynomial with integer coefficients has square-root(2) plus cube-root(3) as a root?" This was new to everyone I showed it to except one very old number theorist who remembered that some really old texts had something about this. Well it turns out that it was the original proof and calculating such examples was a standard exercise over a century ago. However favored proofs have changed and today virtually nobody knows this basic construction! This holds both in the study of symmetric polynomials and algebraic number theory, both originally motivated by the solution to my problem, both having forgotten the solution to my problem, and both having forgotten that they ever had any connection to each other! (The solution is to write in factored form a polynomial which is symmetric in the roots of two other polynomials with integer coefficients. Multiply it out, and after algebraic manipulation you get a new polynomail whose coefficients are polynomials in the coefficients of your original two. The two you start with are "x^2 - 2" and "x^3 - 3". The result you will get, assuming no careless errors, is a 6'th degree polynomial with the desired root.) > - ... and it might be woth investigating how and why ? > - that most mathematicians are not really contributing, > only some of the best ones ? This is unfortunately true. The lesson to be drawn is the importance of keeping lines of communication open and not getting isolated. It isn't that this is an inevitable result. It is quite preventable. But it takes effort. > - ... and that you know how to identify the good ones (or you don't) ? You don't. I have heard many horror stories of very good mathematicians who were polyglots who could not get tenure for the simple reason that while they had a lot of good papers they didn't have enough in any one area to get recommendations. And, of course, nobody on the tenure committee was going to read any of the papers! > - ... and that the other are completely useless ? > or possibly they fulfill another role ? > They fill another role. They teach calculus. This is the main thing that math departments contribute to our society and they are not doing a very good job at it! However I cannot go into that without opening up more cans of worms. Suffice it to say that this is a subject of some controversy in mathematics... Anyways I don't think that mathematics is a very good model for free software to follow. Imagine a community of programmers who are unable to actually try running or compiling a program, who have gone down a social path which reduces the amount of peer review that goes on. This is furthermore a group which (for various reasons) is popularly regarded as dealing with the epitome of pure Truth. Yet I can point at major results that are widely distrusted within mathematics! I am not just talking about mistakes in random papers. I am talking about thinks like the classicification of finite groups (an effort that took decades and is being redone because the people who did it can't read and don't believe their own proof) and the proof of the four color theorem (the debate about which literally comes down to deciding whether the bugs in the program they wrote for their analysis came up when they ran it). I hold free software development to higher standards than that, and I hope that others do as well! Cheers, Ben